Talk:Word problem (mathematics education)

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Cleanup required - 2003
As it was written, this article is childish. All mathematical problems are expressed primarily in words. Any impression to the contrary is found only among those whose acquaintance with mathematics comes only from having been required to learn it from textbooks, and even then it's based only on misperceiving what they've read. A student sees a problem written thus:


 * x2 + 3x + 2

and thinks this is a problem posed in "equations" (a common misnomer among non-mathematicians). Such a student has failed to notice that somewhere above this "equation" there were some words that said "Factor the following polynomials" or "Find the derivative of each of the following functions of x", etc. If those words say "Factor the following polynomials" and one of the polynomials is x2 + 3x + 2 then one problem has been posed; if they say "Find the derivative of each of the following functions of x" then quite a different problem has been posed. That difference sometimes goes unnoticed among those who hate being required to take a math course, and that's a big source of missing the point. Michael Hardy 00:01, 16 Aug 2003 (UTC)

I now see that the only page linking to this one is about the "word problem" in group theory". That problem most certainly is not called the "word problem" because of its being expressed in words! Michael Hardy 00:03, 16 Aug 2003 (UTC)


 * Well, that's my understanding of what word problems are. They're used primarily in high school mathematics. You know, like the ol' "A train is heading blah blah blah", and the whole idea of the question is to test your understanding of the mathematical ideas behind the question...
 * That's a good explanation, I think I'll add it, if you don't mind me unredirecting. There's nothing stopping the article from having a groups explanation as well... Dysprosia 00:18, 16 Aug 2003 (UTC)


 * "Used primarily in high school mathematics"?? That is utter nonsense! As I said, all mathematics problems are expressed in words.  That fact fails to be obvious to students only because they have failed to understand, but it becomes more clear as you get into more advanced work.  Look at Walter Rudin's book Principles of Mathematical Analysis.  Michael Hardy 14:43, 2 Sep 2003 (UTC)


 * This nonsense that there are some mathematical problems that are not expressed primarily in words arises only in the most elementary courses -- through about second-year calculus or so. After that it would become obvious that all mathematical problems are necessarily expressed verbally.  But I suspect that most people who don't figure that out when they are in secondary school or earlier will never reach more advanced courses. Michael Hardy 14:59, 2 Sep 2003 (UTC)


 * Michael, as I see it, there are two outcomes to this issue: either we can get it to an acceptable, NPOV article, or we can leave it as a solid redirect. If we want to go down the first path, sure, that's fine talking about the misconceptions about "word problems", but I don't see what was so wrong about the addition re math. modelling.


 * As it stands, "childish misunderstanding" and "That difference sometimes goes unnoticed among those who hate being required to take a math course" is not NPOV. "The impression to the contrary is found only among those whose acquaintance with mathematics comes only from having been required to learn it from textbooks" is unsubstantiated, and probably inaccurate too (Kids who are taught math in one manner, ie., rote learning of shifting of symbols may get this impression too).


 * If we can't work this out, then I'm happy to leave it as a redirect. But we're both intelligent people, so I'm sure we can! :) Dysprosia 22:42, 2 Sep 2003 (UTC)


 * You began with this:


 * A word problem is sometimes used to describe a mathematical problem posed descriptively

First a grammatical point: The term "word problem" may be used to describe something, but you wrote, not the that term is used to describe something, but that a word problem is used to describe something.


 * However, modelling problems often involve some kind of specific mathematical concept underlying it - these modelling problems are often described entirely in words without needing to resort to "equations".

Why restrict this to "modelling problems"? Where is there any mathematics problem that does not involve specific mathematical concepts? What about this problem: Prove that every metric space that is complete and totally bounded is compact. There is no mathematical notation used, but it's not a "modelling" problem.

(As for the POV problems -- I'll try to find a way to rephrase it while still being realistic.) Michael Hardy 00:18, 3 Sep 2003 (UTC)

I didn't restrict it to modelling problems, I simply added the fact that the term "word problem" could include math modelling questions as well. You are right, such other questions do not involve modelling also and should be added.

As for the grammar, yes, I'm still learning. That is why you're such an asset here! :) Dysprosia 04:45, 3 Sep 2003 (UTC)

Word Problem:

How many drops of rain are falling

at this moment in this city in this storm?

How many

on this patch of asphalt

from   now    to   now?

Count them.

Just commenting
As I read through this article, I could not help but feel that this point: 'No doubt the term "word problem" is sometimes regarded as meaningless' was being driven home rather forcefully. The iteration of the point that "In fact, all mathematical problems are expressed primarily in words", including the use of bold to highlight all makes it feel as though someone has an axe to grind. I am not disputing the point, but the repetition of the statement feels forced. Does anyone have any thoughts on this, (or should I butt out)? Laconic 18:19, 22 Jun 2005 (UTC)


 * I put that there. I have a times been quite irritated by the proposed dichotomy, according to which there are some mathematical problems that are expressed in words and others that are not.  This weird notion is in fact widespread among people who don't get along well with mathematics. Michael Hardy 19:50, 22 Jun 2005 (UTC)


 * While I quite agree with Michael, that all mathematical problems are expressed in a combination of words and mathematical notation. I still think most people "know a word problem when they see it", however (kind of like what the Supreme Court justice said about porn), and it's still possible to come up with a reasonable "definition". I find in my classes, from college algebra to differential equations, students invariably have far more problems with "word problems" than with more computational problems. The distinction to me does not have to do with whether "words" are being used or not. This distinction is that a problem is described verbally in a way that conceals the mathematical expression of the problem. The problem is that students are not accustomed or skilled in translating descriptive statements of a situation into precise mathematical expression. Anyone who has taught college algebra knows what I mean. I believe Davis/Hersh had something to say about this in the Math Experience. 198.59.188.232 02:32, 24 October 2005 (UTC)


 * You're talking only about problems posed as exercises in textbooks, not about math problems generally. And I think you're mistaken in thinking that the problem is primarily with translating words to symbols.  The problem is that when they're in symbols then the student can often solve them without understanding them, by applying algorithms they've been given.  Words, on the other hand, require understanding. Michael Hardy 18:43, 24 October 2005 (UTC)

This misconception is a fairly well known phenomenon in math education with plenty of literature on it. It would be far better to just quote some of this literature and explain the uses and misuses of word problems. Then something like this misconception of some problems being in words and others in equations would be naturally brought up and described in an NPOV manner. Also suitable would be the various theories of what causes these misconceptions and why students may have difficulty with a "word problem". --C S (Talk) 02:22, 20 December 2006 (UTC)

moved from the article:
A Wheelof cheese is sealed in a wax covering. The wheel of cheese is in the shape of a cylinder that has a diameter of 25 centimeters and a height of 20 centimeters. What is the surface area of the cheese that needs to be coveed in wax?24.188.13.108 01:18, 20 December 2006 (UTC)lola

definition
The definition really is quite unbearable at the moment. That's not really what a word problem is, as pointed out by Michael Hardy. I will fix it, and hopefully it will at least be closer to what a lot of educators would call a "word problem". --C S (Talk) 02:26, 20 December 2006 (UTC)


 * This edit] by User:C S makes the definition much worse than it was. If a problem says


 * Find three consecutive odd numbers whose sum is 93.


 * then people call that a "word problem", but it does NOT help students apply mathematics to "real-world situations".
 * The fact is, the very concept of "word problem" is a childish misunderstanding.


 * If it says


 * Differentiate the following functions of x:


 * #1) ......
 * #2) ......
 * #42) x3 &minus; 8x
 * #42) x3 &minus; 8x


 * then students say "It's not a word problem. It only says "x3 &minus; 8x".
 * On the other hand, if it says


 * Factor the following polynomials


 * #1) ......
 * #2) ......
 * #42) x3 &minus; 8x
 * #42) x3 &minus; 8x


 * then again they say "It's not a word problem. It only says "x3 &minus; 8x".
 * But the difference between the two problems is in the words.
 * It does not only say "x3 &minus; 8x".


 * Applying problems to real-world situations has little to do with it. Michael Hardy 19:08, 20 December 2006 (UTC)

There is also some material that says 'The idea is to present mathematics to the students in a less abstract way and to give the students a sense of "usefulness" of mathematics.' That is silly. Again, a problem that says "Find three consecutive odd numbers whose sum is 93" is generally described as a "word problem", but the idea is obviously NOT to give anyone an idea of the usefulness of mathematics. Michael Hardy 19:16, 20 December 2006 (UTC)


 * I'm not sure why you keep bringing up something that everybody understands. Of course words are used in math problems.  From looking at your earliest and latest edits to this article, I get the impression you think the point of this article is to explain "word problem" is a misconception.  But a number of people do use the term "word problem" in a meaningful way.  There appear to be a variety of definitions that are used in the literature.  Some would agree your sample problem is a word problem, while others would not.  I made an attempt to define a type of word problem that is often called a "story problem".  It can definitely be improved upon, but I don't see how what was there before is any better.


 * I also put quotes around "real world", which admittedly wasn't a good way to make the point. But my idea was to convey that many educators do think that kind of problem does help students apply mathematics in the real world, as it involves a kind of translation process that they envision is helpful.


 * I think you should consider that this is something used in math education and thus needs to be explained properly. The article shouldn't be a diatribe on "word problem" not making any sense, but that's how it reads now after your edits.  Additionally, I don't know if your definition is that universal.  I'm going to ask for some second opinions on this.  --C S (Talk) 19:51, 20 December 2006 (UTC)

If that's what you call an example of helping students apply mathematics to the real world, then why aren't things like quadratic equations also classed by you as things intended to help students apply mathematics to the real world. Michael Hardy 21:35, 20 December 2006 (UTC)


 * Uh, why are you asking me? Or are you using a generic "you"?  --C S (Talk) 22:06, 20 December 2006 (UTC)


 * If only to avoid fact, but also for simplicity, I'd leave out any speculation as to why educators think word problems are a good idea. In general we should cut down on the OR and confine the unreferenced claims to uncontroversial facts. If we can find sources describing the educational rationale (which, I suspect, is different from what is stated now), then I'd still propose to delegate any discussion to further on in the article. Clearly, there is a continuum from purely verbal to extremely formally-symbolic problem formulations, with most cases in the wide space between the extremes, a fact that should be made clear but does not need much discussion. This article should concentrate on the verbal side of the spectrum, making clear what is meant by the term. --Lambiam Talk  23:53, 20 December 2006 (UTC)

Where do you find anything that could be suspected of being "OR" in here? I'm baffled. I've been teaching math for a quarter century, and I know no math problems that are stated in extremely formally-symbolic form, with no words. Michael Hardy 02:56, 21 December 2006 (UTC)


 * The following statements are not sourced, are potentially controversial or possibly open to doubt, and can in my opinion be considered OR. (I'm not saying that they are incorrect, but in fact some are clearly a matter of opinion rather than a hard fact.)
 * In fact, all mathematical problems are stated primarily in words
 * but students unskilled in mathematics often fail to realize that.
 * A student looking at #3 may think "It not a word problem; it just says ...
 * On the other hand an exercise that says ... Is considered a "word problem" because the student is not able to be unaware of the problems dependence on verbal expression.
 * It is believed that the first example is useful in helping primary school students to understand the concept of subtraction.
 * The second example, however, might not be so interesting or so "real-life" to a high school student. A high school student may find that it is easier to handle the following problem: ...
 * This type of problem is called a "problem in equations" by some students
 * however the use of "equation" in this sense is sometimes misused to refer to anything at all that is written in mathematical notation.
 * Indeed, in senior high school level or higher, this type of problems is often used solely to test understanding of underlying concepts within a descriptive problem, instead of testing the student's capability to perform algebraic manipulation or other "mechanical" skills. As a result, a word problem may be even harder than the so-called "problems in equations" and indeed, it may inhibit a student's desire to learn mathematics.
 * These problems seem to fail completely in motivating students to learn and therefore are not word problems in the usual sense of the word.
 * it is called a "word problem" just because it is not written as '... !
 * A commonplace misunderstanding of mathematics is that some mathematical problems are expressed primarily in words and others are "problems in equations".
 * People sometimes fail to notice that all mathematical problems are expressed primarily in words.
 * This confusion may result from the way textbooks are written.
 * A student sees this: ... and thinks this is a problem posed in "equations".
 * Worse still, students may consider ... / ...to be two different problems.
 * No doubt the term "word problem" is sometimes regarded as meaningless.
 * --Lambiam Talk 13:40, 21 December 2006 (UTC)

--
 * A student looking at #3 may think "It not a word problem; it just says ...

Hasn't EVERYONE who teaches math run into this one repeatedly?


 * On the other hand an exercise that says ... Is considered a "word problem" because the student is not able to be unaware of the problems dependence on verbal expression.

This one is self-evident: the problem is stated with lots of words and no mathematical notation except for one numeral.


 * however the use of "equation" in this sense is sometimes misused to refer to anything at all that is written in mathematical notation.

Huh??? You've got to be kidding! Not only does EVERYONE who teaches math run into this all the time, but we see it on Wikipedia talk pages incessantly. Michael Hardy 00:47, 22 December 2006 (UTC)


 * Although I'd be happy to talk about my personal experiences, I've never taught elementary maths at what I think must be the level of "junior high". But even if I had, observations I make doing so would also be OR. Some of this may be culture dependent; what do we know about the perceptions of students in Estonia or Malaysia? Students may get confused by any kind of question; if you ask this:
 * ''In each of the following years an important European battle took place; give the name of that battle.
 * '' #1) 9
 * '' #2) 1571
 * '' #3) 1815
 * a student looking at #3 may think "It's not a question; it's just a number" and answer "3×5×112". When I was at the receiving end, word problems were always identified by a heading "Word Problem". The main task was to turn the following piece of prose into a system of equations, and the main difficulty was to make sense of the given word salad. Once you had managed to determine what were the unknowns, identified and extracted the constraints from the verbiage, and compiled the formulas, solving was easy. So to me a word problem is a maths problem in which the main or in any case first task of the solver is to extract a system of equations from a piece of prose that gives constraints between implicit unknowns. If you can read German, please have a look at de:Textaufgabe. --Lambiam Talk  08:07, 22 December 2006 (UTC)

Cleanup 2007
This article, as is now, is more a complaint about the supposed stupidity of the "word problems" concept than an article about them.

There are clearly 2 groups of editors discussing in this talk page, each using a different definition for "word problem":
 * 1) Word problems are problems which need a phase of situation understanding / translation to equations / however you may call it, prior to mechanical calculus.
 * 2) Word problems are problems stated in words instead of using mathematical notation.

The first definition is a subset of the second, and hence the misunderstanding between both sides. The second definition leads to the concept being of no use, and childish, yes.

I believe the correct definition is the first. Although I've never thaugth mathematics to children, I (as everybody) can remember those problems, and I never thought what made them different was the quantity of words. Maybe some students fail to discern them properly and believe that mechanical excercises stated with words are the same thing. Then those students would say "Find the derivative of f(x)." is a word problem while "df(x)/dx = " is not. Well, those students are wrong, but not because both are word problems, but because neither is.

Some people not being able to grasp the concept on sight doesn't make the concept useless. It is a concept used by educators, so to create this article we only need to access published literature directed to math educators and write what is told. It cannot be so difficult. --euyyn 23:50, 15 April 2007 (UTC)


 * Gak, I agree, this article is disgraceful. A serious dose of neutrality is needed. Tagged and added to WP:CLEANUP. -dmmaus 09:46, 15 June 2007 (UTC)


 * There's no great mystery to this. In maths, a "word problem" is simply one where the student must translate a verbal explanation into mathematical notation, rather than having it presented in mathematical notation already. For example, this is a "word problem":


 * "John is twenty years younger than Amy, and in five years' time he will be half her age. What is John's age now?"


 * And this is the equivalent stated as a "non-word" problem:


 * Solve for J:
 * J = A - 20
 * J + 5 = (A + 5)/2


 * Later on the article does give some not dissimilar examples, but the intro is very confusing IMO. I suggest that the article leads off with a simple illustration like the above, rather than tying itself in knots and airing unneeded opinions about what people do or don't realise. Matt 20:02, 16 June 2007 (UTC).


 * I have made an attempt at this cleanup. Much of the article as it was before was dedicated to Michael Hardy's repeated point that "word problem" should mean "a problem that uses words", and so apply to most if not all problems. However much he thinks that this is what "word problem" should mean, it's not what it means. I have rewritten the intro (I took the liberty of using your example, Matt!) and removed the repetitions. Please check that I haven't removed anything important, and if something more could be removed, or should be added. Also check for OR and Weasel traps; there are a lot of "may" and "might" and such in the text that I have found quite hard to dispose of. The thought of finding some secondary sources in the form of literature for math educators is a good one. Anyway, at least we're on our way now. -- Jao 23:31, 23 June 2007 (UTC)
 * I also just ran into this Cleanup. Finding references looks to be key to cleaning it up. Otherwise statements like "The fact that word problems are often considered harder than so-called "problems in equations" could of course lead to the conclusion that they may inhibit a student's desire to learn mathematics rather than provoke it" are just unsupported editorial opinion. Gordonofcartoon 13:27, 28 June 2007 (UTC)
 * I tried to cut out POV opinion material int he artical, tighten it up and make it more presentable. It still needs sources, but should be more suitable now. --Lendorien 12:59, 1 August 2007 (UTC)

Progressive tidy up with sources
I hope I'm not doing too much damage but I'm trying to shift the focus on to published research on word problems. There is vast quantities of it but its hard to summarise and a lot of it is a bit obscure. --Nick Connolly 05:04, 27 September 2007 (UTC) I've removed the unsourced tag - the article still isn't great but it does have sources now. --Nick Connolly 23:45, 27 September 2007 (UTC)

First word problem
it is kind of messy, and doesn't really explain what is happening: "The answer to the word problem is that John is 15 years old, while the answer to the mathematical problem is J = 15 (and A =35)," if you actually follow the math A and J should both equal 10, then when you add 5 to J(10) it equals 15. — Preceding unsigned comment added by 207.47.247.28 (talk) 21:29, 6 July 2011 (UTC)